The new particle allowed by the unified theory at eleven parameters: the second doublet of the empty slot and where to look for it

@misc{reynaud2026particle,
  title = {{The new particle allowed by the unified theory at eleven parameters: the second doublet of the empty slot and where to look for it}},
  author = {Patrick B. Reynaud},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05764070v2},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The new particle allowed by the unified theory at eleven parameters: the second doublet of the empty slot and where to look for it
AU  - Patrick B. Reynaud
PY  - 2026
DA  - 2026/09/25
PB  - HAL
UR  - https://hal.science/hal-05764070v2
AB  - The unified theory at eleven parameters writes the electroweak sector on the hyperquaternion H⊗H⊗H = M 4 (H), the space-time tetraquaternion times the weak isospin, and fills its scalar quaternion with the Higgs doublet. One slot is left empty, the pseudoscalar quaternion ωq 2 , room for a second doublet of opposite intrinsic parity. The particle is permitted, not predicted. This paper works out what the algebra fixes if the slot is filled, and where the particle would show. Since ω commutes with the isospin and squares to -1, Q = q 1 + ωq 2 is one complex quaternion, whose left-handed projection q 1 -iq 2 carries in its two columns the two doublets Φ u,d = (Φ 1 ± iΦ 2 )/ √ 2: a single Yukawa ψQψ is a Type-II two-Higgs-doublet model, flavour-conserving at tree level, with tan β = 1 exactly at the scale where the algebra's statement is made, the vacuum of the scalar slot projecting equally on the two. The parity ω → -ω of the boson sector forbids m 2 12 , λ 6 , λ 7 , allows λ 5 , and makes ⟨Φ 2 ⟩ = 0 an extremum. The alignment is then exact at tree level, the 125 GeV boson keeps its Standard-Model couplings, and the heavy states H, A, H ± have no vacuum value and the couplings of the top Yukawa. The top Yukawa breaks the parity at one loop, and the sizes are computed: tan β runs by three per cent per e-fold and the odd quartics are generated at 0.06 per e-fold, so that the statement is sharp at the electroweak scale and a boundary condition if made at the Planck scale. The sector adds five numbers, a mass m 22 = √ ζ 2 M P and four quartics, and the algebra fixes none of them; the benchmark ζ 2 = ζ, 88.5 GeV, is excluded. Every observable is then a function of a mass and of quartics, with couplings that are fixed. The 2026 rate of b → sγ puts the charged state above about 720 GeV at tan β = 1, the top-pair searches with their interference exclude the neutral pair to about a TeV, and the oblique T bounds the splittings, 32 GeV moving m W by 8 MeV. The diagrams are written out with their expressions and their sizes. For the Higgs boson it is the charged-Higgs triangle in h → γγ, δκ γ = -0.24% λ 3 (800 GeV/m H ± ) 2 , and for the W and the Z their self-energies. For the top it is gg → H, A → t t with its interference, a dip at a TeV where the top loop is absorptive, widths of 48 and 55 GeV and σ(A)/σ(H) = 1.28, and for the charged state the b → sγ penguin and tbH ± production. In the inert reading, in which the Yukawa is ψq 1 ψ alone, it is the inelastic Z exchange of the dark-matter doublet with its kinematic edge, 333 keV at 540 GeV. The top-pair spectrum between 0.8 and 1.5 TeV is where the Type-II reading is found or pushed up, the HL-LHC reaching some 1.2 TeV at tan β = 1 and a 100 TeV collider several TeV. The reading of the LZ event as the inert doublet is excluded by the Sun, as the companion paper recorded, except at the 11-20 TeV of Murayama and Noether; the doublet as dark matter above the solar edge remains, tested by the Higgs portal and the Galactic centre. What would refute the filling as the algebra writes it is listed: an H → hh, A → Zh or t → bH + signal, a flavour change at tree level, or a measured tan β ̸ = 1.
ER  -